A more accurate expression for E osc would be obtained by including additional terms in the TaylorMacLaurin

A more accurate expression for EÌ… _oscwould be obtained by including additional terms in the Taylor–MacLaurin series. The Taylor–MacLaurin series expansion of f (x) in the vicinity of x₀is given by (see Math Supplement)

Use this formalism to better approximate EÌ… _osc by expanding hv/e^{kBT i}n powers of hv/k_BT out to (hv/k_BT)³ in the vicinity of hv/k_BT = 0. Calculate the relative error, EÌ… _osc – kBT/EÌ… _osc. if you had not included the additional terms for ν = 9.00 × 10¹¹ s^−1 at temperatures of 1000., 600., and 200. K.

Transcribed Image Text:

1 d²f(x) 2! dx? 1 d²f(x) df (x) f(x) = f(x) + ( (x – xo)³ + (x – x)² +: 3! (x – xo) + x=X /x=X x=X,